Lemma 3.2 (Second Geometric Lemma, [1]})

There exists a set \(\Lambda _u \subset S^2 \cap \mathbb {Q}^3\)  that consists of vectors \(k\)  with associated orthonormal bases \((k, k_1, k_2)\) , \(\varepsilon _u > 0\) , and smooth positive functions \(\gamma _{(k)}: B_{\varepsilon _u}(\mathrm {Id}) \rightarrow \mathbb {R}\) , where \(B_{\varepsilon _u}(\mathrm {Id})\)  is the ball of radius \(\varepsilon _u\)  centered at the identity in the space of \(3 \times 3\)  symmetric matrices, such that for \(S \in B_{\varepsilon _u}(\mathrm {Id})\)  we have the following identity:
\(S = \sum _{k \in \Lambda _u} \gamma _{(k)}^2(S) k_1 \otimes k_1 .\) 
